## Basic Concepts of Wavelet Transform

Learning Wavelet Transform should start from basic concepts

I published three posts at the very beginning when I started my blogs, which are about visualization of wavelet function and scaling function, visualization of filter bank of a Discrete Wavelet and applications of wavelet transform.

From this post, I will continue to write some practical tutorial articles on how to apply Wavelet Transform to solve real-world problems. It is better to know some basic concepts related with Wavelet Transform (WT) before using it for real-world cases.

### 1. What is a Wavelet?

A wavelet is a wave-like oscillation, which is a “brief oscillation” or “small oscillation”. Its amplitude begins at zero, increases or decreases, and then returns to zero one or more times. Let’s see some wavelets plots as follows:

### 2. Wavelet Transform

In the signal processing context, Wavelet transform (WT), also called wavelet analysis, is a method to decompose an input signal of interest into a set of elementary waveforms, i.e. “wavelets”

WT provides a way to analyze the signal by examining the coefficients (or weights) of these wavelets. It is probably the most recent solution to overcome the shortcomings of the Fourier transform.

#### (1) Two basic factors

• scale (or dilation): Stretched or compressed (or shrunk) wavelet
• shift (or location): the positions of the wavelet

WT uses inner products to compare the similarity between a signal and the wavelet at various scales and positions, i.e. shifted and compressed and stretched versions of a wavelet, to obtain the wavelet coefficients.

#### (2) Scale and frequency

The general correspondence between scale and frequency is:

• Small scale a ⇒ Compressed wavelet ⇒ Rapidly changing details ⇒ High frequency ω.
• Long scale a ⇒ Stretched wavelet ⇒ Slowly changing, coarse features ⇒ Low frequency ω.

### 3. Signal and Time Series

Since WT is a method to deal with an input signal of interest, then what a signal and what different between a signal and a time series.

#### (1) A signal

• It is more related to physics, image analysis, engineering / science domains
• Many signals are basically a time series once they are sampled at a certain time interval, such as speech, audio, power, water and environmental quality data from sensors or labs, etc.
• A signal is more general, not necessary measured by time, maybe spatial coordinates, distances to a source, or multidimensional
• Almost anything carrying information can be interpreted as a signal.

#### (2) Time series

• It is more related to data analysis domain
• Any data points are varied and measured over times, bank interests change, GDP growth, COD changes in river, population growth, etc.
• It usually depicts the data values (y) is a function of time (t): ????=????(????)
• A time series is always indexed by time

In this sense, a time series is a signal varied and measured over times. Then it is not hard to understand why WT is usually applied in time series analysis.

### 4. Signal and Wave

From the above, we know that a signal is a more general and abstract concept, which is almost anything carrying information. A signal has no physical manifestation.

Whereas, a wave is a physical phenomenon, which can be used to transmit signals. Thus, we see a wave as a wave signal. A wave can be modelled and measured.

• Period (????): refers to the time taken to complete one oscillation, i.e. cycle from one peak to the next (or from any point to the next matching point),
• Amplitude (????): the height from the center line to the peak (or to the trough).
• Frequency (????): the number of oscillations per unit time

Besides, a wave has length, called wavelength, which is the distance over which a wave repeats, which can be defined as the distance between two successive crests or troughs of a wave.

In addition, waves typically has different forms, called waveform, which describes the shape or form of a wave signal. There are four basic types of waveform, such as

• Sine wave
• Square wave
• Triangle wave
• Sawtooth wave

### 5. Online courses

If you are interested in learning how to apply wavelet transform to real-world cases with Python step by step, you are welcome to enroll my courses:

#### (1) Very basic one

Practical Python Wavelet Transforms (I): Fundamentals

#### (2) For Real-world projects

Practical Python Wavelet Transforms (II): 1D DWT

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